SFB 823
Weak convergence of the
empirical copula process with respect to weighted metrics
Discussion Paper
Betina Berghaus, Axel Bücher,Stanislav Volgushev
Nr. 38/2014
Weak convergence of the empirical copula process with respect to weighted metrics
Betina Berghaus, Axel B¨ucher and Stanislav Volgushev∗ November 21, 2014
Abstract
The empirical copula process plays a central role in the asymptotic analysis of many statistical procedures which are based on copulas or ranks. Among other applications, results regarding its weak con- vergence can be used to develop asymptotic theory for estimators of dependence measures or copula densities, they allow to derive tests for stochastic independence or specific copula structures, or they may serve as a fundamental tool for the analysis of multivariate rank statistics.
In the present paper, we establish weak convergence of the empirical copula process (for observations that are allowed to be serially depen- dent) with respect to weighted supremum distances. The usefulness of our results is illustrated by applications to general bivariate rank statistics and to estimation procedures for the Pickands dependence function arising in multivariate extreme-value theory.
Keywords and Phrases: Empirical copula process; weighted weak conver- gence; strongly mixing; bivariate rank statistics; Pickands dependence func- tion.
AMS Subject Classification: 62G30, 60F17.
1 Introduction
The theory of weak convergence of empirical processes can be regarded as one of the most powerful tools in mathematical statistics. Through the continuous mapping theorem or the functional delta method, it greatly fa- cilitates the development of asymptotic theory in a vast variety of situations (Van der Vaart and Wellner,1996).
∗Ruhr-Universit¨at Bochum, Fakult¨at f¨ur Mathematik, Universit¨atsstr. 150, 44780 Bochum, Germany. E-mail: betina.berghaus@rub.de, axel.buecher@rub.de, stanislav.volgushev@rub.de. This work has been supported by the Collaborative Research Center “Statistical modeling of nonlinear dynamic processes” (SFB 823) of the German Research Foundation (DFG) which is gratefully acknowledged.
For applying the continuous mapping theorem or the functional delta method, the course of action is often similar. Consider for instance the con- tinuous mapping theorem: starting from some abstract weak convergence result, say Fn F in some metric space (D, dD), one would like to deduce weak convergence of φ(Fn) φ(F), where φ is some mapping defined on (D, dD) with values in another metric space (E, dE). This conclusion is pos- sible provided φ is continuous at every point of a set which contains the limitF, almost surely (Van der Vaart and Wellner,1996).
The continuity ofφis linked to the strength of the metricdD– a stronger metric will make more functions continuous. For example, letD=`∞([0,1]) denote the space of bounded functions on [0,1] and consider the real-valued functional φ(f) := R
(0,1)f(x)/x dx (with φ defined on a suitable subspace ofD). In Section3.2below, this functional will turn out to be of great inter- est for the estimation of Pickands dependence function and it is also closely related to the classical Anderson-Darling statistic. Now, if we equipDwith the supremum distance, as is typically done in empirical process theory, the map φ is not continuous because 1/x is not integrable. Continuity of φ can be ensured by considering a weighted distance, such as for instance supx∈[0,1]|f1(x) −f2(x)|/g(x) for a positive weight function g such that g(x)/x is integrable. Similar phenomenas arise with the functional delta method, seeBeutner and Z¨ahle(2010). It thus is desirable to establish weak convergence results with the metricdDtaken as strong as possible. One class of metrics which is of particular interest in many statistical applications is given by weighted supremum distances.
For classical empirical processes, corresponding weak convergence results are well known. For example, the standardd-dimensional empirical process Fn(x) = √
n{Fn(x)−F(x)} with F having standard uniform marginals, converges weakly with respect to the metric induced by the weighted norm
kGkω = sup
u∈[0,1]d
G(u) {g(u)}ω
, g(u) = mind
j=1uj
∧ 1−mind
j=1 uj ,
ω∈(0,1/2). See, e.g.,Shorack and Wellner (1986) for the one-dimensional i.i.d.-case, Shao and Yu (1996) for the one-dimensional time series case or Genest and Segers(2009) for the bivariate i.i.d.-case. For d= 2, the graph of the function gis depicted in Figure 1.
The present paper is motivated by the apparent lack of such results for the empirical copula process ˆCn. This process, an element ofD([0,1]d) precisely defined in Section 2 below, plays a crucial role in the asymptotic analysis of statistical procedures which are based on copulas or ranks. Unweighted weak convergence of ˆCn has been investigated by several authors under a variety of assumptions on the smoothness of the copula and on the temporal dependence of the underlying observations, seeGaenssler and Stute(1987);
Fermanian et al.(2004);Segers(2012);B¨ucher and Volgushev(2013);B¨ucher
0.0 0.2
0.4 0.6
0.8 1.0
0.0 0.2 0.4 0.6 0.8 0.01.0 0.20.1 0.3 0.4 0.5
0.0 0.2
0.4 0.6
0.8 1.0
0.0 0.2 0.4 0.6 0.8 0.01.0 0.20.1 0.3 0.4 0.5
Figure 1: Graphs of g(u, v) = min{u, v,1−min(u, v)} (left picture) and of
˜g(u, v) = min{u, v,(1−u),(1−v)} (right picture).
et al. (2014), among others. However, results regarding its weighted weak convergence are almost non-existent. To the best of our knowledge, the only reference appears to be R¨uschendorf (1976), where, however, weight functions are only allowed to approach zero at the lower boundary of the unit cube. The restrictiveness of this condition becomes particularly visible in dimensiond= 2 where it is known that the limit of the empirical copula process is zero on the entire boundary of the unit square (Genest and Segers, 2010). This observation suggests that, for d = 2, it should be possible to maintain weak convergence of the empirical copula process when dividing by functions of the form {˜g(u, v)}ω where
˜
g(u, v) =u∧v∧(1−u)∧(1−v).
A picture of the graph of ˜g can be found in Figure 1, obviously, we have
˜
g≤g. The main result of this paper confirms the last-mentioned conjecture.
More precisely, we establish weighted weak convergence of the empirical cop- ula process in general dimensiond≥2 with weight functions that approach zero wherever the potential limit approaches zero. We also do not require the observations to be i.i.d. and allow for exponential alpha mixing.
Potential applications of the new weighted weak convergence results are extensive. As a direct corollary, one can derive the asymptotic behavior of Anderson-Darling type goodness-of-fit statistics for copulas. The deriva- tion of the asymptotic behavior of rank-based estimators for the Pickands dependence functions (Genest and Segers, 2009) can be greatly simplified and, moreover, can be simply extended to time series observations. Through a suitable partial integration formula, the results can also be exploited to derive weak convergence of multivariate rank statistics as for instance of certain scalar measures of (serial) dependence. The latter two applications are worked out in detail in Section3 of this paper.
The remaining part of this paper is organized as follows. In Section2, the empirical copula process is introduced and the main result of the paper, its weighted weak convergence, is stated. In Section3, the main result is illus- tratively exploited to derive the asymptotics of multivariate rank statistics and of common estimators for extreme-value copulas. All proofs are deferred to Section 4, with some auxiliary results postponed to Section 5. Finally, AppendixAin the supplementary material contains some general results on (locally) bounded variation and integration for two-variate functions which are needed for the proof of Theorem3.3.
2 Weighted empirical copula processes
Let X = (X1, . . . , Xd)0 be a d-dimensional random vector with joint cu- mulative distribution function (c.d.f.) F and continuous marginal c.d.f.s F1, . . . , Fd. The copula C of F, or, equivalently, the copula of X, is de- fined as the c.d.f. of the random vector U = (U1, . . . , Ud)0 that arises from marginal application of the probability integral transform, i.e.,Uj =Fj(Xj) for j = 1, . . . , d. By construction, the marginal c.d.f.s of C are standard uniform on [0,1]. By Sklar’s Theorem, C is the unique function for which we have
F(x1, . . . , xd) =C{F1(x1), . . . , Fd(xd)}
for all x= (x1, . . . , xd)∈Rd.
Let Xi, i = 1, . . . , n be an observed stretch of a strictly stationary time series such thatXi is equal in distribution toX. SetUi = (Ui1, . . . , Uid)∼ C with Uij = Fj(Xij). Define (observable) pseudo observations ˆUi = ( ˆUi1, . . . ,Uˆid) of C through ˆUij = nFnj(Xij)(/n+ 1) for i = 1, . . . , n and j= 1, . . . , d. The empirical copula ˆCn of the sampleX1, . . . ,Xn is defined as the empirical distribution function of ˆU1, . . . ,Uˆn, i.e.,
Cˆn(u) = 1 n
n
X
i=1
1( ˆUi≤u), u∈[0,1]d. The corresponding empirical copula process is defined as
u7→Cˆn(u) =√
n{Cˆn(u)−C(u)}.
Forω≥0, define a weight function
gω(u) = min{∧dj=1uj,∧dj=1[1−(u1∧ · · · ∧ubj∧ · · · ∧ud)]}ω,
where the hat-notation u1∧ · · · ∧ubj ∧ · · · ∧ud is used as a shorthand for min{u1, . . . , uj−1, uj+1, . . . , ud}. For d= 2, the function is particularly nice and reduces togω(u1, u2) = min(u1, u2,1−u1,1−u2)ω, see Figure1. Note that for vectorsu∈[0,1]dsuch that at least one coordinate is equal to 0 or
such that d−1 coordinates are equal to 1, we have gω(u) = 0. As already mentioned in the introduction for the case d= 2, these vectors are exactly the points where the limit of the empirical copula process is equal to 0, almost surely, whence one might hope to obtain a weak convergence result forCn/gω. To prove such a result, a smoothness condition onC has to be imposed.
Condition 2.1. For every j ∈ {1, . . . , d}, the first oder partial derivative C˙j(u) := ∂C(u)/∂uj exists and is continuous on Vj = {u ∈ [0,1]d : uj ∈ (0,1)}. For every j2, j2 ∈ {1, . . . , d}, the second order partial derivative C¨j1j2(u) := ∂2C(u)/∂uj1∂uj2 exists and is continuous on Vj1 ∩Vj2. More- over, there exists a constantK >0 such that
|C¨j1j2(u)| ≤Kmin
1
uj1(1−uj1), 1 uj2(1−uj2)
, ∀u∈Vj1 ∩Vj2.
For completeness, define ˙Cj(u) = lim suph→0{C(u+hej)−C(u)}/hwher- ever it does not exist. Note, that Condition2.1coincides with Condition 2.1 and Condition 4.1 in Segers(2012), who used it to prove Stute’s represen- tation of an almost sure remainder term (Stute, 1984). The condition is satisfied for many commonly occurring copulas (Segers,2012).
For−∞ ≤ a < b≤ ∞, let Fab denote the sigma-field generated by those Xi for which i∈ {a, a+ 1, . . . , b}and define, for k≥1,
α[X](k) = sup
|P(A∩B)−P(A)P(B)|:A∈ F−∞i , B∈ Fi+k∞ , i∈Z as the alpha-mixing coefficient of the time series (Xi)i∈Z. The sequence is called strongly mixing (or alpha-mixing) ifα[X](k)→0 fork→ ∞. Finally,
αn(u) =√
n{Gn(u)−C(u)}, Gn(u) =n−1Pn
i=11(Ui ≤u), denotes the (unobservable) empirical process based onU1, . . . ,Un.
Theorem 2.2. (Weighted weak convergence of the empirical copula process)Suppose that X1,X2, . . . is a stationary, alpha-mixing sequence with α[X](k) = O(ak), as k → ∞, for some a∈ (0,1). If the marginals of the stationary distribution are continuous and if the corresponding copulaC satisfies Condition 2.1, then, for any c∈(0,1) and any ω∈(0,1/2),
sup
u∈[c
n,1−c
n]d
Cˆn(u)
gω(u) −C¯n(u) gω(u)
=oP(1)
where, for anyu∈[0,1]d,
C¯n(u) :=αn(u)−
d
X
j=1
C˙j(u)αn(u(j)),
with u(j) = (1, . . . ,1, uj,1, . . . ,1). Moreover, we have C¯n/˜gω CC/˜gω in (`∞([0,1]d),k · k∞), where g˜ω(u) =gω(u) +1{gω(u) = 0}, where
CC(u) =αC(u)−
d
X
j=1
C˙j(u)αC(u(j)),
and where αC denotes a tight, centered Gaussian process with covariance Cov{αC(u), αC(v)}=X
i∈Z
Cov{1(U0 ≤u),1(Ui ≤v)}.
The proof of Theorem 2.2is given in Section4.1below. In fact, we state a more general result which is based on conditions on the usual empirical processαn. These conditions are subsequently shown to be valid for expo- nentially alpha-mixing time series.
3 Applications
Theorem 2.2 may be exploited in numerous ways. For instance, many of the most powerful goodness-of-fit tests for copulas are based on distances between the empirical copula and a parametric estimator for C (Genest et al.,2009). The results of Theorem2.2can be exploited to validate tests for a richer class of distances, as for weighted Kolomogorov-Smirnov orL2-dis- tances. Second, estimators for extreme-value copulas can often be expressed through improper integrals involving the empirical copula (see Genest and Segers,2009, among others). Weighted weak convergence as in Theorem2.2 facilitates the anlysis of their asymptotic behavior and allows to extend the available results to time series observations. Details regarding the CFG- and the Pickands estimator are worked out in Section3.2below.
Theorem2.2may also be used outside the genuine copula framework, for instance, for proving asymptotic normality of multivariate rank statistics.
The power of that approach lies in the fact that proofs for time series are essentially the same as for i.i.d. data sets. In Section3.1, we derive a general weak convergence result for bivariate rank statistics.
3.1 Bivariate rank statistics
Bivariate rank statistics constitute an important class of real-valued statis- tics that can be written as
Rn= 1 n
n
X
i=1
J( ˆUi1,Uˆi2)
for some function J : (0,1)2 → R, called score function. Rn can also be expressed as a Lebesgue-Stieltjes integral with respect to ˆCn, i.e.,
Rn= Z
[n+11 ,n+1n ]2
J(u, v)d ˆCn(u, v),
which offers the way to derive the asymptotic behavior of Rn from the asymptotic behavior of the empirical copula. This idea has already been exploited inFermanian et al.(2004): however, in their Theorem 6,J has to be a bounded function which is not the case for many interesting examples.
Also, the uniform central limit theorems for multivariate rank statistics in van der Vaart and Wellner(2007) require rather strong smoothness assump- tions on J (which imply boundedness ofJ).
Example 3.1. (Rank Autocorrelation Coefficients)SupposeY1, . . . , Yn
are drawn from a stationary, univariate time series (Yi)i∈Z. Rank autocor- relation coefficients of lag k∈Nare statistics of the form
rn,k= 1 n−k
n
X
i=k+1
J1
n n
n+1Fn(Yi) o
J2
n n
n+1Fn(Yi−k) o
,
whereJ1, J2are real-valued functions on (0,1) andFn denotes the empirical cdf ofY1, . . . , Yn. For example, the van der Waerden autocorrelation (Hallin and Puri,1988) is given by
rn,k,vdW = 1 n−k
n
X
i=k+1
Φ−1 n n
n+1Fn(Yi) o
Φ−1 n n
n+1Fn(Yi−k) o
,
(with Φ and Φ−1 denoting the cdf of the standard normal distribution and its inverse, respectively) and the Wilcoxon autocorrelation (Hallin and Puri, 1988) is defined as
rn,k,W = 1 n−k
n
X
i=k+1
n n
n+1Fn(Yi)−1 2
o
logn n+1n Fn(Yi−k) 1−n+1n Fn(Yi−k)
o .
Obviously, the corresponding score functions are unbounded. Asymptotic normality for these and similar rank statistics has been shown for i.i.d. ob- servations and for ARM A-processes (Hallin et al., 1985). To the best of our knowledge, no general tool to handle the asymptotic behavior of such statistics for dependent observations seems to be available. Theorem 3.3 below aims at partially filling that gap.
Example 3.2. (The pseudo-maximum likelihood estimator)As a common practice in bivariate copula modeling one assumes to observe a sampleX1, . . . ,Xnfrom a bivariate distribution whose copula belongs to a parametric copula family, parametrized by a finite-dimensional parameter θ∈Θ⊂Rp. Except for the assumption of absolute continuity, the marginal distributions are often left unspecified in order to allow for maximal ro- bustness with respect to potential miss-specification. In such a setting, the
pseudo-maximum likelihood estimator (seeGenest et al.(1995) for a theoret- ical investigation) provides the most common estimator for the parameterθ.
Ifcθ denotes the corresponding copula density, the estimator is defined as θˆn= arg maxθ∈Θ
n
X
i=1
log{cθ( ˆUi1,Uˆi2)}.
Using standard arguments from maximum-likelihood theory and imposing suitable regularity conditions, the asymptotic distribution of √
n(ˆθn−θ0) can be derived from the asymptotic behavior of
Rn= 1 n
n
X
i=1
Jθ0( ˆUi1,Uˆi2), (3.1) whereθ0denotes the unknown true parameter and whereJθ = (∂logcθ)/(∂θ) denote the score function. Typically, this function is unbounded, as for in- stance in case of the bivariate Gaussian copula model whereθ is the corre- lation coefficient and the score function takes the form
Jθ(u, v) = θ(1−θ2)−θ{Φ−1(u)2+ Φ−1(v)2}+ (1 +θ2)Φ−1(u)Φ−1(v)
1 +θ2 .
Still, the conditions of Theorem3.3below can be shown to be valid.
Finally, note that pseudo-maximum likelihood estimators also arise in Markovian copula models (Chen and Fan,2006) where copulas are used to model the serial dependence of a stationary time series at lag one. Again, their asymptotic distribution may be derived from rank statistics as in (3.1).
The following theorem is the central result of this section. It establishes weak convergence of bivariate rank-statistics by exploiting weighted weak convergence of the empirical copula process. For the definition of the space of functions of locally bounded total variation in the sense of Hardy-Krause, BV HKloc((0,1)2), and for Lebesgue-Stieltjes integrals with respect to such functions, we refer the reader to DefinitionA.8in the supplementary mate- rial. The proof is given in Section4.4.
Theorem 3.3. Suppose the conditions of Theorem2.2are met. Moreover, suppose thatJ ∈BV HKloc((0,1)2) is right-continuous and that there exists ω∈(0,1/2)such that |J(u)| ≤const×gω(u)−1 and such that
Z
(0,1)2
gω(u)|dJ(u)|<∞. (3.2) Moreover, for δ→0, suppose that
Z
(δ,1−δ]
|J(du, δ)|=O(δ−ω) and Z
(δ,1−δ]
|J(du,1−δ)|=O(δ−ω), (3.3) Z
(δ,1−δ]
|J(δ,dv)|=O(δ−ω) and Z
(δ,1−δ]
|J(1−δ,dv)|=O(δ−ω). (3.4)
Then, asn→ ∞,
√n{Rn−E[J(U)]}
Z
(0,1)2
CC(u)dJ(u).
The weak limit is normally distributed with mean0 and variance σ2 =
Z
(0,1)2
Z
(0,1)2
E[CC(u)CC(v)]dJ(u)dJ(v).
Remark 3.4. (i) Provided the second order partial derivative ¨J12(u, v) :=
∂2J(u, v)/∂u∂v exists, then the conditions (3.2)–(3.4) are equivalent to R
(0,1)2gω(u, v)|J¨12(u, v)|d(u, v)<∞ and, as δ→0, Z 1−δ
δ
|J˙1(u, δ)|du=O(δ−ω) and Z 1−δ
δ
|J˙1(u,1−δ)|du=O(δ−ω), Z 1−δ
δ
|J˙2(δ, v)|dv=O(δ−ω) and Z 1−δ
δ
|J˙2(1−δ, v)|dv=O(δ−ω), where ˙J1(u, v) :=∂J(u, v)/∂u,J˙2(u, v) :=∂J(u, v)/∂v.
(ii) A careful check of the proof of Theorem 3.3 shows that the theorem actually remains valid under the more general conditions of Theorem 4.5 below, withω∈(0,1/2) replaced byω ∈(0,2(1−θθ1
1) ∧2(1−θθ2
2)∧(θ3−1/2)).
As a simple application of Theorem3.3let us return to the autocorrelation coefficients from Example3.1. It can easily be shown that bothJvdW(u, v) = Φ−1(u)Φ−1(v) andJW(u, v) = (u−12) log(1−vv ) satisfy the conditions of The- orem 3.3. To prove this forJvdW use that |Φ−1(u)| ≤ {u(1−u)}−ε for any ε >0 and that φ{Φ−11(u)} ≤ {u(1−u)}−1, withφdenoting the density of the standard normal distribution. Therefore, both coefficients are asymptot- ically normally distributed for any stationary, exponentially alpha-mixing time series provided that the copula of (Yt, Yt−k) satisfies Condition 2.1.
This broadens results from Hallin et al. (1985), which may be further ex- tended along the lines of Remark 3.4(ii) by a more thorough investigation of Conditions4.1–4.3. Details are omitted for the sake of brevity.
3.2 Nonparametric estimation of Pickands dependence func- tion
Theorem 2.2 can be used to extend recent results for the estimation of Pickands dependence functions. Recall that C is a multivariate extreme- value copula if and only if C has a representation of the form
C(u) = exp
Xd
j=1
loguj
A logu1
Pd
j=1loguj, . . . , logud−1
Pd
j=1loguj
, u∈(0,1)d,
for some functionA: ∆d−1→[1/d,1], where ∆d−1 denotes the unit simplex
∆d−1 ={w= (w1, . . . , wd−1) ∈[0,1]d−1 :Pd−1
j=1wj ≤1}. In that case, A is necessarily convex and satisfies the relationship
max(w1, . . . , wd)≤A(w1, . . . , wd−1)≤1 (wd= 1−Pd−1 j=1wj), for all w ∈ ∆d−1. By reference to Pickands (1981), A is called Pickands dependence function. Nonparametric estimation methods forA in the i.i.d.
case and under the additional assumption that the marginal distributions are known have been considered inPickands(1981);Deheuvels(1991);Cap´era`a et al. (1997); Jim´enez et al. (2001), among others. In the more realistic case of unknown marginal distribution, rank-based estimators have for in- stance been investigated inGenest and Segers (2009);B¨ucher et al. (2011);
Gudendorf and Segers (2012); Berghaus et al. (2013), among others. For illustrative purposes, we restrict attention to the rank-based versions of the Pickands estimator in Gudendorf and Segers (2012) in the following, even though the results easily carry over to, for instance, the CFG-estimator.
The Pickands-estimator is defined as AˆPn(w) =
"
1 n
n
X
i=1
min
n−log( ˆUi1)
w1 , . . . ,−log( ˆUid) wd
o
#−1
and it follows by simple algebra (see Lemma 1 in Gudendorf and Segers, 2012) that APn :=√
n( ˆAPn −A) =−A2BPn/(1 +n1/2BPn),where BPn(w) =
Z 1 0
Cˆn(uw1, . . . , uwd)du u .
Note that R1
0 u−1du does not converge, which hinders a direct application of the continuous mapping theorem to deduce weak convergence ofBPn (and hence of APn) in `∞(∆d−1) just on the basis of (unweighted) weak conver- gence of ˆCn. Deeper results are necessary and in fact, Genest and Segers (2009) and Gudendorf and Segers (2012) deduce weak convergence of BPn
by using Stute’s representation for the empirical copula process based on i.i.d. observations (see Stute, 1984; Tsukahara, 2005) and by exploiting a weighted weak convergence result forαn.
With Theorem2.2, we can give a much simpler proof. Write BPn(w) =
Z 1 0
Cˆn(uw1, . . . , uwd) min(uw1, . . . , uwd)ω
min(uw1, . . . , uwd)ω
u du.
Then, since R1
0 min(uw1, . . . , uwd)ω duu ≤R1
0 uω/d−1du exists for any ω > 0, weak convergence ofBPn is a direct consequence of the continuous mapping theorem and Theorem2.2. Note that this method of proof is not restricted to the i.i.d. case.
4 Proofs
4.1 Proof of Theorem 2.2
Theorem 2.2 will be proved by an application of a more general result on the empirical copula process. For its formulation, we need a couple of ad- ditional conditions which, subsequently, will be shown to be satisfied for exponentially alpha-mixing time series.
Condition 4.1. There exists someθ1 ∈(0,1/2] such that, for allµ∈(0, θ1) and all sequences δn→0, we have
Mn(δn, µ) := sup
|u−v|≤δn
|αn(u)−αn(v)|
|u−v|µ∨n−µ =oP(1).
Condition 4.1 can for instance be verified in the i.i.d. case with θ1 = 1/2, exploiting a bound for the multivariate oscillation modulus derived in Proposition A.1 inSegers (2012).
Condition 4.2. The empirical processαn converges weakly in `∞([0,1]d) to some limit processαC which has continuous sample paths, almost surely.
For i.i.d. samples, the latter condition is satisfies with αC being a C- Brownian bridge, i.e., a centered Gaussian process with continuous sample paths, a.s., and with Cov{αC(u), αC(v)}=C(u∧v)−C(u)C(v).
Condition 4.3. There exist θ2 ∈ (0,1/2] and θ3 ∈ (1/2,1] such that, for any ω∈(0, θ2), anyλ∈(0, θ3) and all j= 1, . . . , d, we have
sup
uj∈(0,1)
αnj(uj) uωj(1−uj)ω
=OP(1), sup
uj∈(1/nλ,1−1/nλ)
βnj(uj) uωj(1−uj)ω
=OP(1), whereαnj(uj) =√
n{Gnj(uj)−uj}and βnj(uj) =√
n{G−nj(uj)−uj}.
Here,Gnj(uj) =n−1Pn
i=11(Uij ≤uj) and, for a distribution functionH on the reals, H− denotes the (left-continuous) generalized inverse function ofH defined as
H−(u) := inf{x∈R:H(x)≥u}, 0< u≤1,
and H−(0) = sup{x∈R:H(x) = 0}. In the i.i.d. case, Condition 4.3 is a mere consequence of results inCs¨org˝o et al.(1986), withθ2= 1/2 , θ3 = 1.
The following proposition shows that the (probabilistic) Conditions 4.1, 4.2and 4.3are satisfied for sequences that are exponentially alpha-mixing.
Proposition 4.4. Suppose that X1,X2, . . . is a stationary, alpha-mixing sequence with α[X](k) = O(ak), as k → ∞, for some a ∈ (0,1). Then, Conditions 4.1, 4.2and 4.3 are satisfied withθ1 =θ2= 1/2 and θ3 = 1.
Here, Condition 4.3 is a mere consequence of results in Shao and Yu (1996) and Cs¨org˝o and Yu (1996), whereas Condition 4.2 has been shown inRio (2000). For the proof of Condition 4.1, we can rely on results from Kley et al.(2014). The precise arguments are given in Section 4.2below.
The following theorem can be regarded as a generalization of Theorem2.2:
weighted weak convergence of the empirical copula process takes place pro- vided the abstract Conditions4.1, 4.2 and 4.3 are met. The proof is given in Section4.3below.
Theorem 4.5. (Weighted weak convergence of empirical copula processes)Suppose Conditions 2.1, 4.1 and 4.3 are met. Then, for any c∈(0,1)and any ω ∈(0,2(1−θθ1
1) ∧2(1−θθ2
2)∧(θ3−1/2)), sup
u∈[nc,1−nc]d
Cˆn(u)
gω(u) −C¯n(u) gω(u)
=oP(1).
If additionally Condition4.2is met, then C¯n/˜gω CC/˜gω in(`∞([0,1]d),k·
k∞).
Proof of Theorem 2.2. The theorem is a mere consequence of Proposition4.4 and Theorem 4.5.
4.2 Proof of Proposition 4.4
For an r-dimensional random vector (Y1, . . . Yr)0, define the rth order joint cumulant by
cum(Y1, . . . Yr) = X
{ν1,...,νp}
(−1)p−1(p−1)!E Y
j∈ν1
Yj
× · · · ×E Y
j∈νp
Yj ,
where the summation extends over all partitions{ν1, . . . , νp},p∈ {1, . . . , r}, of {1, . . . , r}. The following lemma will be one of the main tools for estab- lishing Condition4.1under exponentially alpha-mixing.
Lemma 4.6. If Y1, Y2, . . . is a strictly stationary sequence of random vari- ables with|Yi| ≤K <∞ and if there exist constants ρ∈(0,1) andK0 <∞ such that for any p∈N and arbitrary i1, . . . , ip∈Z
|cum(Yi1, . . . , Yip)| ≤K0ρmaxk,`|ik−i`|,
then, there exist constants C1, C2 < ∞ only depending on K, K0 and |νr| such that
cum
Xn
i=1
Yi, j ∈νr
≤C1(n+ 1)ε(|logε|+ 1)C2, where ε=E[|Yi|].
